Gravity action on rapidly varying metrics

We consider a four-dimensional simplicial complex and the minisuperspace general relativity system on it. The metric is flat in most parts of the interior of every 4-simplex, with the exception of a thin layer of thickness \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\propto \varepsilon}$$\end{document} along each three-dimensional face. In this layer the metric undergoes a jump between the two 4-simplices sharing this face. At \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon \to 0}$$\end{document} this jump would become a discontinuity. However, a discontinuity of the metric induced on the face is not allowed in general relativity: terms arise in the Einstein action tending to infinity as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon \to 0}$$\end{document} . In the path integral approach, these terms lead to the pre-exponent factor with δ-functions requiring that the metric induced on the faces be continuous. That is, the 4-simplices fit on their common faces. The other part of the path integral measure corresponds to the action, which is the sum of independent terms over the 4-simplices. Therefore this part of the path integral measure is the product of independent measures over the 4-simplices. The result obtained is in accordance with our previous one obtained from symmetry considerations.